Abstract

We suggest overcoming the “Rayleigh catastrophe” and reaching superresolution for imaging with both spatially and temporally correlated field of a superradiant quantum antenna. Considering far-field radiation of two interacting spontaneously emitting two-level systems, we show that for the measurement of the temporally delayed second-order correlation function of the scattered field, the Fisher information does not tend to zero with diminishing the distance between a pair of scatterers even for non-sharp time-averaged detection. For position estimation of more scatterers, the measurement of the time-delayed function is able to provide a considerable accuracy gain over the zero-delayed function. We also show that the superresolution with the considered quantum antenna can be achieved for both near-field imaging and for estimating the antenna parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (6)

M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99(1), 012305 (2019).
[Crossref]

S. Zhou and L. Jiang, “Modern description of Rayleigh’s criterion,” Phys. Rev. A 99(1), 013808 (2019).
[Crossref]

R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, and D. Oron, “Super-resolution enhancement by quantum image scanning microscopy,” Nat. Photonics 13(2), 116–122 (2019).
[Crossref]

D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13(2), 80–90 (2019).
[Crossref]

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, “Quantum-enhanced noise radar,” Appl. Phys. Lett. 114(11), 112601 (2019).
[Crossref]

Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, “Quantum-limited estimation of the axial separation of two incoherent point sources,” Optica 6(5), 534–541 (2019).
[Crossref]

2018 (5)

R. E. Evans, M. K. Bhaskar, D. D. Sukachev, C. T. Nguyen, A. Sipahigil, M. J. Burek, B. Machielse, G. H. Zhang, A. S. Zibrov, E. Bielejec, H. Park, M. Loncar, and M. D. Lukin, “Photon-mediated interactions between quantum emitters in a diamond nanocavity,” Science 362(6415), 662–665 (2018).
[Crossref]

V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, “Tailoring nonlinear processes for quantum optics with pulsed temporal-mode encodings,” Optica 5(5), 534–550 (2018).
[Crossref]

A. Mikhalychev, D. Mogilevtsev, G. Y. Slepyan, I. Karuseichyk, G. Buchs, D. L. Boiko, and A. Boag, “Synthesis of quantum antennas for shaping field correlations,” Phys. Rev. Appl. 9(2), 024021 (2018).
[Crossref]

S. Pirandola, R. B. Bhaskar, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” Nat. Photonics 12(12), 724–733 (2018).
[Crossref]

J. Řeháček, Z. Hradil, D. Koutnỳ, J. Grover, A. Krzic, and L. Sánchez-Soto, “Optimal measurements for quantum spatial superresolution,” Phys. Rev. A 98(1), 012103 (2018).
[Crossref]

2017 (6)

J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. Sánchez-Soto, “Optimal measurements for resolution beyond the Rayleigh limit,” Opt. Lett. 42(2), 231–234 (2017).
[Crossref]

J. Řehaček, Z. Hradil, B. Stoklasa, M. Paúr, J. Grover, A. Krzic, and L. Sánchez-Soto, “Multiparameter quantum metrology of incoherent point sources: towards realistic superresolution,” Phys. Rev. A 96(6), 062107 (2017).
[Crossref]

W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118(7), 070801 (2017).
[Crossref]

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95(6), 063847 (2017).
[Crossref]

A. Classen, J. von Zanthier, M. Scully, and G. S. Agarwal, “Superresolution via structured illumination quantum correlation microscopy,” Optica 4(6), 580–587 (2017).
[Crossref]

M. J. Padgett and R. W. Boyd, “An introduction to ghost imaging: quantum and classical,” Philos. Trans. R. Soc., A 375(2099), 20160233 (2017).
[Crossref]

2016 (6)

M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, “Achieving the ultimate optical resolution,” Optica 3(10), 1144–1147 (2016).
[Crossref]

C. Lupo and S. Pirandola, “Ultimate precision bound of quantum and sub-wavelength imaging,” Phys. Rev. Lett. 117(19), 190802 (2016).
[Crossref]

R. Nair and M. Tsang, “Far-field superresolution of thermal electromagnetic sources at the quantum limit,” Phys. Rev. Lett. 117(19), 190801 (2016).
[Crossref]

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6(3), 031033 (2016).
[Crossref]

F. Yang, A. Tashchilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” Optica 3(10), 1148–1152 (2016).
[Crossref]

Z. S. Tang, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” Opt. Express 24(19), 22004–22012 (2016).
[Crossref]

2015 (4)

M. Tsang, “Quantum limits to optical point-source localization,” Optica 2(7), 646–653 (2015).
[Crossref]

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref]

Z. Zhang, S. Mouradian, F. N. Wong, and J. H. Shapiro, “Entanglement-enhanced sensing in a lossy and noisy environment,” Phys. Rev. Lett. 114(11), 110506 (2015).
[Crossref]

B. Brecht, D. V. Reddy, C. Silberhorn, and M. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5(4), 041017 (2015).
[Crossref]

2014 (1)

E. Wolfe and S. Yelin, “Certifying separability in symmetric mixed states of n qubits, and superradiance,” Phys. Rev. Lett. 112(14), 140402 (2014).
[Crossref]

2013 (2)

E. Lopaeva, I. R. Berchera, I. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110(15), 153603 (2013).
[Crossref]

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

2010 (1)

F. Ferri, D. Magatti, L. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref]

2009 (1)

S. Guha and B. I. Erkmen, “Gaussian-state quantum-illumination receivers for target detection,” Phys. Rev. A 80(5), 052310 (2009).
[Crossref]

2008 (2)

S. Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science 321(5895), 1463–1465 (2008).
[Crossref]

S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, “Quantum illumination with Gaussian states,” Phys. Rev. Lett. 101(25), 253601 (2008).
[Crossref]

2007 (1)

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13(4), IFH1 (2007).
[Crossref]

2004 (3)

R. Tanaś and Z. Ficek, “Entangling two atoms via spontaneous emission,” J. Opt. B: Quantum Semiclassical Opt. 6(3), S90–S97 (2004).
[Crossref]

Y. C. Eldar, “Minimum variance in biased estimation: bounds and asymptotically optimal estimators,” IEEE Trans. Signal Process. 52(7), 1915–1930 (2004).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref]

1999 (1)

L. Guosui, G. Hong, and S. Weimin, “Development of random signal radars,” IEEE Trans. Aerosp. Electron. Syst. 35(3), 770–777 (1999).
[Crossref]

1995 (1)

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref]

1988 (1)

Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats in intensity correlations of spontaneous emission from two non-identical atoms,” J. Mod. Opt. 35(1), 81–91 (1988).
[Crossref]

1987 (1)

Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Phys. A (Amsterdam, Neth.) 146(3), 452–482 (1987).
[Crossref]

1970 (1)

1879 (1)

L. Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
[Crossref]

Agarwal, G. S.

Allen, E. H.

E. H. Allen and M. Karageorgis, “Radar systems and methods using entangled quantum particles,” (2008). US Patent 7,375,802.

Ang, S. Z.

S. Z. Ang, R. Nair, and M. Tsang, “Quantum limit for two-dimensional resolution of two incoherent optical point sources,” Phys. Rev. A 95(6), 063847 (2017).
[Crossref]

Ansari, V.

Balaji, B.

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, “Quantum-enhanced noise radar,” Appl. Phys. Lett. 114(11), 112601 (2019).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref]

Berchera, I. R.

E. Lopaeva, I. R. Berchera, I. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110(15), 153603 (2013).
[Crossref]

Bhaskar, M. K.

R. E. Evans, M. K. Bhaskar, D. D. Sukachev, C. T. Nguyen, A. Sipahigil, M. J. Burek, B. Machielse, G. H. Zhang, A. S. Zibrov, E. Bielejec, H. Park, M. Loncar, and M. D. Lukin, “Photon-mediated interactions between quantum emitters in a diamond nanocavity,” Science 362(6415), 662–665 (2018).
[Crossref]

Bhaskar, R. B.

S. Pirandola, R. B. Bhaskar, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” Nat. Photonics 12(12), 724–733 (2018).
[Crossref]

Bielejec, E.

R. E. Evans, M. K. Bhaskar, D. D. Sukachev, C. T. Nguyen, A. Sipahigil, M. J. Burek, B. Machielse, G. H. Zhang, A. S. Zibrov, E. Bielejec, H. Park, M. Loncar, and M. D. Lukin, “Photon-mediated interactions between quantum emitters in a diamond nanocavity,” Science 362(6415), 662–665 (2018).
[Crossref]

Boag, A.

A. Mikhalychev, D. Mogilevtsev, G. Y. Slepyan, I. Karuseichyk, G. Buchs, D. L. Boiko, and A. Boag, “Synthesis of quantum antennas for shaping field correlations,” Phys. Rev. Appl. 9(2), 024021 (2018).
[Crossref]

Boiko, D. L.

A. Mikhalychev, D. Mogilevtsev, G. Y. Slepyan, I. Karuseichyk, G. Buchs, D. L. Boiko, and A. Boag, “Synthesis of quantum antennas for shaping field correlations,” Phys. Rev. Appl. 9(2), 024021 (2018).
[Crossref]

Bourassa, J.

C. W. S. Chang, A. M. Vadiraj, J. Bourassa, B. Balaji, and C. M. Wilson, “Quantum-enhanced noise radar,” Appl. Phys. Lett. 114(11), 112601 (2019).
[Crossref]

Boyd, R. W.

Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, “Quantum-limited estimation of the axial separation of two incoherent point sources,” Optica 6(5), 534–541 (2019).
[Crossref]

M. J. Padgett and R. W. Boyd, “An introduction to ghost imaging: quantum and classical,” Philos. Trans. R. Soc., A 375(2099), 20160233 (2017).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref]

Brecht, B.

V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, “Tailoring nonlinear processes for quantum optics with pulsed temporal-mode encodings,” Optica 5(5), 534–550 (2018).
[Crossref]

B. Brecht, D. V. Reddy, C. Silberhorn, and M. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5(4), 041017 (2015).
[Crossref]

Breuer, H.-P.

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).

Brida, G.

E. Lopaeva, I. R. Berchera, I. Degiovanni, S. Olivares, G. Brida, and M. Genovese, “Experimental realization of quantum illumination,” Phys. Rev. Lett. 110(15), 153603 (2013).
[Crossref]

Buchs, G.

A. Mikhalychev, D. Mogilevtsev, G. Y. Slepyan, I. Karuseichyk, G. Buchs, D. L. Boiko, and A. Boag, “Synthesis of quantum antennas for shaping field correlations,” Phys. Rev. Appl. 9(2), 024021 (2018).
[Crossref]

Burek, M. J.

R. E. Evans, M. K. Bhaskar, D. D. Sukachev, C. T. Nguyen, A. Sipahigil, M. J. Burek, B. Machielse, G. H. Zhang, A. S. Zibrov, E. Bielejec, H. Park, M. Loncar, and M. D. Lukin, “Photon-mediated interactions between quantum emitters in a diamond nanocavity,” Science 362(6415), 662–665 (2018).
[Crossref]

Capmany, J.

D. Marpaung, J. Yao, and J. Capmany, “Integrated microwave photonics,” Nat. Photonics 13(2), 80–90 (2019).
[Crossref]

Chang, C. W. S.

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Figures (6)

Fig. 1.
Fig. 1. (a) The scheme of the simplest quantum antenna for producing correlated photon pairs registered in the far-field zone with two detectors seen at the angles $\theta _1$ and $\theta _2$ from the $x$ axis. The angle $\delta \alpha$ denotes antenna rotation. (b) The Fisher information for detecting the rotation angle $\delta \alpha$ in dependence on the direction angle $\theta$ is as given by Eq. (15) for the delay $\Gamma \tau =0.75$. Solid, dashed, dash-dotted and dotted lines correspond to the following averaging times: $\Gamma \delta \tau =0,0.25,0.5,0.75$. For this picture the distance between dipoles is $k\lambda /2\pi =1$.
Fig. 2.
Fig. 2. The scheme of the far-field scattering imaging with the simplest quantum antenna. The field produced by the antenna in the far-field zone impinges on set of targets, and the scattered field is detected in the far-field zone by two detectors, ${\mathrm D_1}$ and ${\mathrm D_2}$.
Fig. 3.
Fig. 3. An illustration of the reconstruction of the distance between scatterers for the zero-delay measurements (a) and for the delayed $G^{(2)}$ measurements in dependence on this distance. Dashed lines depict the lower bound on errors as predicted by the Fisher information for $N=10^7$ counts per measurement. The red line is the guide for the eye showing true separation between objects. For both panels $\zeta _{12}=4.8$, the angle between axes of the antenna and direction to the object is 1.2 rad, observation angles for the first and second measurements are $\pi /2.15$ and $\pi /2.15 + \pi /50$. For the panel (b) delays for the first and second measurements are $\Gamma \tau =0.5,0.75$ respectively. The Fisher information was calculated for the probabilities normalized by their sum.
Fig. 4.
Fig. 4. Inset on the panel (c) shows the three-scatterers system, $r_1$ and $r_2$ are distances to be inferred. The total error bound defined by the trace of the inverse Fisher information matrix is shown for different values of the parameter $kr$, for the zero-delay measurements (a) and the delayed measurements (b). The panel (c) shows the values of the total error on a slice $r_1=r_2/2=r$ for different values of the parameter $kr$, for the zero-delay measurements (dashed line) and the delayed measurements (solid line). The position of the slice is shown in the panels (a) and (b). For all the panels $\zeta _{12}=4.8$; the angle between axis of the antenna and the object is 1.2 rad; two detectors for the same observation angle were considered for $\phi _1=\phi _2=\pi /4,\pi /3,\pi /2,3\pi /4$. For the delayed measurements the following delays were taken $\Gamma \tau =1.5,2.5,2,0$. The Fisher information was calculated for the probabilities normalized by their sum.
Fig. 5.
Fig. 5. (a) The Fisher information for detection the distance between scatterers with a single-angle measurement for different delays and angles $\theta$ between axes of the antenna and the object. in dependence on the distance between TLS. Solid, dotted, and dashed lines correspond to the following delays and observation angles $\Gamma \tau =0.75,\theta =\pi /4$, $\Gamma \tau =1.0,\theta =\pi /3$, $\Gamma \tau =1.5,\theta =\pi /3$. (b) The Fisher information for scattered far-field detection in dependence on the inter-dipole distance, $\zeta _{12}$. Gray and black lines correspond to the averaging time $\Gamma \delta \tau =0$ and 0.05. For all panels the delay $\Gamma \tau =0.75$.
Fig. 6.
Fig. 6. The scheme of the near-field imaging with the simplest quantum antenna. The field produced by the antenna impinges on the two small holes in the object plane and propagates toward the image plane. The direction angles toward these holes are $\theta +\delta \psi$ and $\theta$. This plane is in the near-field zone with respect to the object plane, the function $h({\vec S},{\vec S}_I)$ describes the field propagation between planes. Then, the field is detected by two detectors, $S_0$ are distances between the lens and the image plane, and between the object plane and the lens.

Equations (22)

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F k l = j 1 p j ( x k p j ( x ; y ) ) ( x l p j ( x ; y ) ) .
Δ k 2 1 N [ F 1 ] k k .
Δ 2 = k = 1 M Δ k 2 1 N T r [ F 1 ] 1 N f m i n ,
E ( r , t ` ) j = 1 2 n × [ n × d ] | r | exp { i ω c r | r | R j } σ j ( t ) ,
d d t ρ = i f 12 [ σ 1 + σ 2 + h . c . , ρ ] + 1 2 j , l = 1 2 γ j l ( 2 σ j ρ σ l + σ j + σ l ρ ρ σ j + σ l ) ,
f 12 = 3 2 Γ ( cos { ζ 12 } ζ 12 ( sin { ζ 12 } ζ 12 2 + cos { ζ 12 } ζ 12 3 ) ) .
γ 11 = γ 22 = Γ , γ 12 = 3 2 Γ ( sin { ζ 12 } ζ 12 ( sin { ζ 12 } ζ 12 3 cos { ζ 12 } ζ 12 2 ) ) ,
E o u t ( r , t ` ) = j = 1 , 2 f j ( r , x ) σ j ( t ) ,
G ( 2 ) ( θ 1 , t ` ; θ 2 , t ` + τ ) = j , l , m , n = 1 2 F j , l , m , n ( θ 1 , θ 2 ) σ j + ( t ) σ l + ( t + τ ) σ m ( t + τ ) σ n ( t ) ,
σ j + ( t ) σ l + ( t + τ ) σ m ( t + τ ) σ n ( t ) = 1 2 exp { Γ ( 2 t + τ ) } Υ j n l m ( τ ) ,
Υ j n j n ( τ ) = cos { f 12 τ } + cosh { γ 12 τ } ,
Υ j n n j ( τ ) = Υ j j n n ( τ ) = cos { f 12 τ } + cosh { γ 12 τ } , Υ j n j j ( τ ) = Υ j j j n ( τ ) = i sin { f 12 τ } sinh { γ 12 τ } , Υ j j n j ( τ ) = Υ n j j j ( τ ) = i sin { f 12 τ } sinh { γ 12 τ } .
E ( θ i , t ` ) = f 1 σ 1 ( t ) + f 2 σ 2 ( t ) ,
p ( θ , δ α ; τ ) = D exp { Γ τ } [ ( 1 + cos 2 ϕ ( θ + δ α ) ) cosh { γ 12 τ } 2 cos ϕ ( θ + δ α ) sinh { γ 12 τ } + sin 2 ϕ ( θ + δ α ) cos { f 12 τ } ] ,
F D p ¯ ( θ , δ α ; τ ) ( δ α p ¯ ( θ , δ α ; τ ) ) 2 , p ¯ ( θ , δ α ; τ ) = τ δ τ τ + δ τ d x Ω ( x ) p ( θ , δ α ; x ) ,
E ( ϕ l , t ´ ) exp { i k R s } R s ( E ( r 1 , t ` ) + E ( r 2 , t ` ) exp { i d k cos ϕ l } ) f 1 ( ϕ l ) σ 1 ( t ) + f 2 ( ϕ l ) σ 2 ( t ) ,
f 1 ( ϕ l ) 1 + exp { i k R o δ α cos ϕ l } , f 2 ( ϕ l ) exp { i ζ 12 cos θ } + exp { i ζ 12 cos { θ + δ α } } exp { i k R o δ α cos ϕ l } .
f 2 ( π / 2 ) exp { i ζ 12 cos θ } + exp { i ζ 12 cos { θ + δ α } } exp { i ζ 12 cos θ } ( 2 + i ζ 12 δ α sin { θ } ) 2 exp { i ζ 12 cos { θ + δ α / 2 } } .
p ( θ , δ α ; 0 ) = D | f 1 ( ϕ 1 ) f 2 ( ϕ 2 ) + f 1 ( ϕ 2 ) f 2 ( ϕ 1 ) | 2 .
E ( ϕ l , t ´ ) exp { i k R s } R s j = 1 3 E ( r j , t ` ) exp { i d j 1 k cos ϕ l } ,
E I ( S I , t ) = O d 2 S h ( S I , S ) E ( S , t ) o ( h ( S I , S 1 ) E ( S 1 , t ) + h ( S I , S 2 ) E ( S 2 , t ) ) = f 1 ( S I ) σ 1 ( t ) + f 2 ( S I ) σ 2 ( t ) ,
f 1 1 + P S F ( δ ψ ) , f 2 exp { i ζ 12 cos θ } + exp { i ζ 12 cos { θ + δ ψ } } P S F ( δ ψ ) ,

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